2.2 Counting Outcomes
Overview of the lesson plan:
Is it necessary to play a game many times to figure out the likelihood of
winning? If not, how can we measure this likelihood? In this lesson, students
learn that they can find the probability of winning a game by finding the ratio
all the possible ways to win to all the possible outcomes for the game.
Daily goals:
Students will
1. Learn to calculate how many “tickets” there are by using tree diagrams
to count.
2. Identify winning “tickets” on the tree diagram, understanding that
there are multiple possible combinations to make a winning ticket.
3. Represent the probability as a fraction based on the tree diagram.
Summary of activities:
Introduction to counting outcomes and probability: Teacher uses
the single-bet roulette example in Chances Part I to introduce the
concept of counting outcomes and writing probability as a fraction.
Creating a probability tree: Students create a probability tree to
represent all possible outcomes in the Color Pick game. Students
identify and count winning “tickets” on the tree to come up with the
probability of winning the game.
Mathematical ideas
This lesson introduces students to the idea of finding probability by counting
outcomes. Students explore the meaning of what an outcome is and learn to
create a tree diagram to count the outcomes in order to find the probability.
Making a model: In this lesson, students learn about the importance of
having a systematic way to count all possible outcomes in a combination or
permutation situation. The tree diagram is one specific tool that students
learn to use in this lesson. The tree diagram provides a way for students to
visually organize the possible outcomes so that there is less chance of
missing an outcome. Some students may already have their own way of
counting systematically; if so, help them make connections with the tree
diagram so that they will have multiple tools for counting outcomes in the
future.
Materials:
PowerPoint for Chances 2
Color branches and leaves in labeled bags
Chart paper, 1 for each group
Glue stick, 1 for each group
Scissors, 1 for each group
Tape, 1 for each group
2.2 Counting Outcomes Lesson Plan
City Digits
Lesson Plan Outline
12:00pm Introduction to probability (10 min)
Do now (Slide 2)
The Do Now question serves as a
warm up to remind students about
the previous lesson and the factors
one must consider to measure
probability.
Go over homework – have students share their game ideas and
explanations
Introducing probability (Slide 3)
Answers vary. This question is
meant to be open-ended and
encourage students to talk about
what probability means and explain
their sense of how 1 in 38
describes probability.
Slide 4
Use this as an opportunity to bring
up what happened yesterday and
ask Orlando to reiterate his point
that if the chances were closer to
50% then there would have been
more winners.
12:10pm Make a Model (10 min)
Slide 5
Ask students to come up with all the ways
that you could have drawn a winner in the
color-pick game.
Before you go to the tree model, give
students a chance to think about this
question. They might try to start making
a list, but it gets hard to make a list in a
systematic way.
2.2 Counting Outcomes Lesson Plan
City Digits
Begin to model the use of a tree diagram
to organize possible outcomes. The idea is
that rather than randomly coming up with
a list of all the possible combinations, we
can go about it systematically.
12:20pm Work time (20 min)
Slide 7. Give students instructions for this activity:
Break up students into groups. Each group should get a set of color
branches–one color for each of the colors used in the simulation.
Circulate and help students as they work.
Students get their leaves after they have arranged their branches
and demonstrate that they understand what their leaves should look
like.
Take a photo of each group’s branch to make a collective tree on
the smart board.
12:40pm Counting leaves (5 min)
Counting all leaves: Slide 8
[gather photos of each group’s branch to create class tree diagram]
[gather photos of
each group’s
branch to create
class tree
diagram]
12:45pm
Q: Is there a way to figure this out without
counting each leaf?
Encourage them to use their tree diagrams to
explain. The purpose of this is to have
students look across all the branches and see
that they are the same in structure.
Students should see that there are 5 ways to
choose the first color. Then for each of these
5 ways, you can pair them with 4 other
colors, then for each of the 20 pairs, we have
3 choices for the third color, so we multiply
that by 3, or 5 x 4 x 3.
Finding Probability (10 min)
Slide 9
Choose/draw a winning combination.
Have student bring up winning leaves to
the board.
Discuss idea that each trio can be
arranged 6 different ways. In other words, each
combination can be arranged 6 different ways.
Now let’s explore how many distinct
combinations there are.
2.2 Counting Outcomes Lesson Plan
City Digits
You can have students identify and group together as a class by
moving around the leaves and grouping them. The result should be
10 groups of 6 leaves each.
[Alternatively, have students do the grouping first and THEN discuss
what happens if when we are choosing a winner.]
Slide 10
Students should see that when a trio
of colors is drawn for the color pick game,
there are 6 distinct “leaves” or outcomes
that could be winners out of 60 total
outcomes. This can be reduced to 1/10.
Another way to see this is as 1 of 10
leaf groups. Since we agreed that order does
not matter, we might consider each group of
leaves as an outcome. In this case, then there is 1 group, or 1
outcome, out of 10 groups, or 10 outcomes, that would be the
winner.
From this illustration, students should realize that it is important to
be consistent in thinking about how one is defining outcomes, but as
long as consistency is upheld, the answer should still be the same.
12:55
Closing
If time, this can launch a
mini-discussion on
estimating and comparing
fractions.
Exit Ticket/Homework
Slide 12
2.2 Counting Outcomes Lesson Plan
City Digits